href="#fb3_img_img_8d765dcf-63b4-5489-8a88-fdd8e356d08b.jpg" alt="image"/> as the greatest integer k, such that φ is in For any
PROPOSITION 1.9.– We have the inclusion:
[1.25]
and moreover, the metric space endowed with the distance defined by [1.24] is complete.
PROOF.– Take any x ∈ ℋp+q − 1, and any
Recall that we denote by |x| the minimal n, such that x ∈ ℋn. Since |x1| + |x2| = |x| ≤ p + q — 1, either |x1| ≤ p – 1 or |x2| ≤ q — 1, so the expression vanishes. Now, if (ψn) is a Cauchy sequence in
□
As a corollary, the Lie algebra
1.3.5. Characters
Let ℋ be a connected filtered Hopf algebra over k, and let A be a k-algebra. We will consider unital algebra morphisms from ℋ to the target algebra
The notion of character involves only the algebra structure of ℋ. On the contrary, the convolution product on
PROPOSITION 1.10.– Let ℋ be any Hopf algebra over k, and let be a commutative k-algebra. Then, the characters from ℋ to form a group under the convolution product, and for any , the inverse is given by:
[1.26]
PROOF.– Using the fact that Δ is an algebra morphism, we have for any x, y ∈ ℋ:
If
The unit
We call infinitesimal characters with values in the algebra those elements α of
PROPOSITION 1.11.– Let ℋ be a connected filtered Hopf algebra, and suppose that is a commutative algebra. Let (respectively ) be the set of characters of ℋ with values in (respectively the set of infinitesimal characters of ℋ with values in ). Then, is a subgroup of G, the exponential restricts to a bijection from onto , and is a Lie subalgebra of .
PROOF.– Take two infinitesimal characters α and β with values in
Using the commutativity of
which shows that
as easily seen by induction on n. A straightforward computation then yields:
with
The series above makes sense thanks to connectedness, as explained in section 1.3.4. Now let
